[SOLVED] Energy Band Model and Energy bond Model

[Robotduck]

1. Does Bond model really specifies the exact position of the defects, or missing atom in a ( for example Si crystal ) ?

2. In the formation of energy bands, does the energy states overlap just before splitting into conduction band and valence band ?
and why do electrons want to occupy the lowest energy states all the times no matter which atom or molecule you look at ?
Also when forming energy bands , do we take sp3 hybridised orbitals or s and p orbitals separately into account ?
Why cannot we have He molecule ? Is it because all the states are filled in s orbital ?

Thank you in advance !!

 

[c_mitra]

1. No; energy band model is obtained for a crystal with no defects. Defects are introduced as perturbations. Defects are randomly placed and individual placements are of no consequence.

2. The splitting of the energy levels are due to "exchange integral" or resonance process which is due to the fact that electrons are indistinguishable.

3. At low temp, the electrons will always go into the lowest available energy state (this makes the system lower in total energy); at 0K, the energy of the system should be the lowest.

4. In simple text books, the individual orbitals are not shown separately but if you look up journals, you will find them all the way...

5. For He molecule and atom are same; you mean why we do not have He2 molecule, right? That is because 1s and 2s energy levels are VERY different. Energy of a structure with 1s2 and 2s2 both filled is too high (this is a crude description) compared to two separate atoms.

 

[Robotduck]

2.The splitting of the energy levels are due to "exchange integral" or resonance process which is due to the fact that electrons are indistinguishable.

According to Paulis exclusion principle, the electrons are distinguishable. Right ? I do not follow that.
Also, Si bonds with other atoms via sp3 hybridisation, means you have 4 sp3 orbitals. But before forming sp3 orbitals, we have one s and two p orbitals which are filled. My question is when forming energy bands- do hybridised orbitals ( sp3) participate or one s and 2 p ( both filled and empty states ) ?
Also when the energy bands are formed: Do all the discrete energy levels come from the valence shell of different Si atoms? .. are these discrete levels - s and p orbitals of the different Si atoms ?

Thank you

 

[c_mitra]

According to Paulis exclusion principle, the electrons are distinguishable. Right ? I do not follow that.

Electrons are labelled by their quantum numbers. All electrons in an atom has different quantum numbers. (quantum numbers in a molecule are labelled differently: core and valence electrons).

But when we exchange two electrons we get the same energy (e1e2 and e2e1) are degenerate configurations. That is what is responsible for the exchange integral (or resonance energy).

See the derivation of Fermi-Dirac statistics; the electron labelling and the degenerate energy levels will be somewhat clearer (don't worry about the mathematics; just read the arguments)

Also, Si bonds with other atoms via sp3 hybridisation, means you have 4 sp3 orbitals. But before forming sp3 orbitals, we have one s and two p orbitals which are filled

You will have 2 electrons in 3s level and 2 electrons in 3p levels. A total of 4 electrons. After hybridization, you will have four sp3 orbital (that are degenerate) and each will have one electron (before bonding).

In the crystal, all the four sp3 hybrid orbitals form bonds with other sp3 hybrid orbitals. Then the valence band is completely full. (what happens to the empty d orbitals?)

It is not possible for me to explain the real band structure of Si crystal in detail.

Just a hint: one sp3 orbital of one atom interacts with another sp3 orbital of another atom (two centre bond) will give two energy levels- one bonding and another anti-bonding. These antibonding orbitals produce the conduction band (that are completely empty at abs zero temp)